Two-dimensional bilayer blue phosphorus Dirac-like material: a multi-orbital tight-binding investigation

Abstract

This study presents a theoretical examination of the electronic band structure of AA (AB) stacked bilayer blue phosphorus system within the fifth intralayer (5NN) and second interlayer nearest-neighbor (2NN) multi-orbital tight-binding (MOTB) approach. The variation of energy levels has been investigated through the symmetrical tensile strain of the low-buckled honeycomb lattice. Here, the primary objective is to examine the existence of Dirac electronic features in hexagonal stacked bilayer geometry. Our theoretical calculations predict that the AA bilayer is a new hexagonal two-dimensional material with p_x,y-orbital Dirac-like states at the high-symmetry K point. Consequently, these systems can host massless (massive) Dirac fermions. In particular, the AA bilayer exhibits zero-gap Dirac-like properties and manifests distinguishable Dirac-like cones in the presence of weak spin–orbit coupling when a modest stretch of 2.30% is achieved with a remarkably high Fermi velocity of approximately v_f ≈ 0.12 × 10⁵ m s⁻¹. The behavior of the dispersion bands aligns reasonably well with recent experimental observations. Moreover, a stretch of 7.17% breaks some of the sublattice equivalence and enhances the spin–orbit interaction, resulting in the emergence of an electronic band gap of approximately ≈ 0.27 eV in the proximity of the high-symmetry K point. Furthermore, the tiny gap induced by the spin–orbit interaction implies topological nontriviality in the electronic state (quantum anomalous Hall state) of the honeycomb lattice. These findings categorize the AA bilayer as a rare two-dimensional Dirac-like material. This work provides, to the extent of our knowledge, a pioneering investigation into the existence of Dirac electronic properties in bilayer blue phosphorus. In addition, we present the first derivation of the MOTB model. However, the identified electronic characteristics designate this two-dimensional system as an ideal candidate for high-performance nanoelectronic devices.